The AP is certainly within their rights in wanting to protect
their copyright. A
New
York Times blog entry
explains the history of some subtleties that are often not mentioned
in discussions of fair use. However, the AP seems to be going considerably
what that history would permit.

Consider this statement (from the afore-cited NY Times article)
by an AP executive: "… it is more appropriate for
blogs to use short summaries of A.P. articles rather than direct
quotations, even short ones." Direct quotation is almost certainly within
the scope of fair use;
a summary of the doctrine
by the U.S. Copyright Office itself cites a government report as
permitting "summary of an address or article, with brief quotations, in a
news report". The same report notes that "quotation of excerpts in a
review or criticism for purposes of illustration or comment" is
permissible. Where is the violation?

It gets worse. The
Making
Light
blog has a link to an apparent
copyright
license page
that wants to charge you $12.50 for quoting 5-25 words on a web site,
and with
terms
and conditions
that prohibit using licensed quotations to say nasty things about the AP.
That's certainly within their rights as a matter of contract law, but
you almost certainly don't need a copyright license to quote them
if you're criticizing them. Again, to quote the Copyright Office's web
site, courts have regarded "quotation of excerpts in a review or criticism
for purposes of illustration or comment" as within the scope of fair use.
(I should note that I wrote "apparent" above because I haven't found
a link to that page from an AP news story, except on that site.
Also note that that web page existed before this controversy.)

If you're interested, the "fair use" provisions of U.S. copyright law
are given in
17 USC 107.
However, the exact scope is deliberately not defined by the statute;
one has to rely on case law.

Presumably, the AP thinks that it's losing money because of these
excerpts, and they'd make more money if bloggers simply published
links. But you need some context to induce people to click on the
link, such as an excerpt. Besides, links are generally transient.
When I reference AP wire stories on this blog, I generally use
the links via Google;
here
is the AP story on copyright issue.
But a month from now, that story will be gone
and the link will be broken.
If the actual text is relevant to my point, what should I do? For that
matter, if a reader wishes to check the original story for more details
or to see if I misrepresented the story, how should that be done?
(Should the AP sell a permalink generator?
Perhaps I could buy a right to generate some number of permalinks
per month to their stories. I'd certainly consider paying for that.)

The other possibility is that it's about control. Try this one: think
back to your favorite quotation from someone who wishes he or she
really hadn't said that. You want to publish some criticism of that
— but the offending party informs you that the statement was part of
a copyrighted performance that you don't have the right to reuse.
Preposterous, right? But that's being tried!
Prof. Wendy Seltzer
used a
clip of the NFL's
copyright statement to teach her class about the limits of copyright.
Naturally, she received a DMCA takedown notice. She replied that her
posting fell within the fair use provisions of the law, so
YouTube
reinstated the video. This,
of course,
resulted
in another takedown notice from the NFL.
Cluelessness? Or an attempt to silence a critic?